Problem: Simplify the following expression: $y = \dfrac{5x^2- 22x+21}{5x - 7}$
Explanation: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(21)} &=& 105 \\ {a} + {b} &=& &=& {-22} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $105$ and add them together. The factors that add up to ${-22}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-7}$ and ${b}$ is ${-15}$ $ \begin{eqnarray} {ab} &=& ({-7})({-15}) &=& 105 \\ {a} + {b} &=& {-7} + {-15} &=& -22 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({5}x^2 {-7}x) + ({-15}x +{21}) $ Factor out the common factors: $ x(5x - 7) - 3(5x - 7)$ Now factor out $(5x - 7)$ $ (5x - 7)(x - 3)$ The original expression can therefore be written: $ \dfrac{(5x - 7)(x - 3)}{5x - 7}$ We are dividing by $5x - 7$ , so $5x - 7 \neq 0$ Therefore, $x \neq \frac{7}{5}$ This leaves us with $x - 3; x \neq \frac{7}{5}$.